Consider the Problem a Finite Difference Approximation of the Solution Is Desired

Multiple Choice

Consider the problem $\frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } = 0 , u ( 0 , y ) = 0 , u ( x , 0 ) = 0 , u ( 1 , y ) = y - y ^ { 2 } , u ( x , 1 ) = x - x ^ { 2 }$ . A finite difference approximation of the solution is desired, using the approximation of the previous problem. Use a mesh size of $h = 1 / 3$ The conditions satisfied by the mesh points on the boundary are Select all that apply.

A) $u = 0$ at (0, 1/3) and (1/3, 0)
B) $u = 0$ at (0, 2/3) and (2/3, 0)
C) $u = 0$ at (1/3, 1/3) and (2/3, 2/3)
D) $u = 2 / 9$ at (1, 1/3) and (1/3, 1)
E) $u = 2 / 3$ at (1, 2/3) and (2/3, 1)